math

source/3D Location Encoder/NeRF.md

@@ -36,6 +36,15 @@ The `NERFSpatialRelationLocationEncoder` is designed to compute spatial embeddin
     <img src="../images/NeRF.png" alt="NeRF-transformation" title="NeRF-transformation" width="60%" />
 </p>
 
+Encoded \( x \) = \(\bigoplus_{i=0}^{L-1}\) \([ \sin(2^i \pi x), \cos(2^i \pi x)]\)
+
+Encoded \( y \) = \(\bigoplus_{i=0}^{L-1}\) \([ \sin(2^i \pi y), \cos(2^i \pi y)]\)
+
+Encoded \( z \) = \(\bigoplus_{i=0}^{L-1}\) \([ \sin(2^i \pi z), \cos(2^i \pi z)]\)
+
+Where ⊕ denotes concatenation of vectors.
+
+
 ### Configuration Parameters
 - **coord_dim**: Dimensionality of the space being encoded (e.g., 2D, 3D).
 - **frequency_num**: Number of different sinusoidal frequencies used to encode spatial differences.

source/3D Location Encoder/xyz.md

@@ -46,9 +46,18 @@ Processes a batch of coordinates and converts them into spatial relation embeddi
 - **Formulas:**
   - Convert latitude `lat` and longitude `lon` coordinates into radians.
   - Calculate `x, y, z` coordinates using the following equations:
-    <p align="center">
-      <img src="https://drive.google.com/uc?id=1tviSk-NbxB0G8fTx5vYpkvBJXmiBK08g" alt="xyz-transformation" title="xyz-transformation" width="80%" />
+
+      $$x = \cos(lat) \times \cos(lon)$$  
+      $$y = \cos(lat) \times \sin(lon)$$  
+      $$z = \sin(lat)$$
     </p>
+
+    Where:
+
+    - *lat* is the latitude coordinate in radians.
+    - *lon* is the longitude coordinate in radians.
+    - *x*, *y*, *z* are the resulting Cartesian coordinates.
+
   - Concatenate `x, y, z` coordinates to form the high-dimensional vector representation.
 
 - **Returns:**

source/Basic Concepts/Single point location encoder.md

@@ -9,6 +9,10 @@ output:
   <img src="../images/single_location_encoder_structure.png" alt="Location Encoder Structure" title="General Structure of Single Location Encoder" width="30%" />
 </p>
 
+$$
+Enc(\mathbf{x}) = \mathbf{NN}(PE(\mathbf{x}))
+$$
+
 
 ## EncoderMultiLayerFeedForwardNN()  
 `NN(⋅) : ℝ^W -> ℝ^d` is a learnable neural network component which maps the input position embedding `PE(x) ∈ ℝ^W` into the location embedding `Enc(x) ∈ ℝ^d`. A common practice is to define `NN(⋅)` as a multi-layer perceptron, while Mac Aodha et al. (2019) adopted a more complex `NN(⋅)` which includes an initial fully connected layer, followed by a series of residual blocks. The purpose of `NN(⋅)` is to provide a learnable component for the location encoder, which captures the complex interaction between input locations and target labels.